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A sports coach intends to choose a team of players from a pool of candidates. The coach wants to be able to have more than 20 but fewer than 25 distinct possibilities for the composition of the chosen team, with at least as many candidates chosen for the team as those not chosen.

Identify the number of candidates in the pool and the number of players on the team that are consistent with the coach’s intentions. Make only two selections, one in each column.

A sports coach intends to choose a team of players from a pool of candidates. The coach wants to be able to have more than 20 but fewer than 25 distinct possibilities for the composition of the chosen team, with at least as many candidates chosen for the team as those not chosen.

Identify the number of candidates in the pool and the number of players on the team that are consistent with the coach’s intentions. Make only two selections, one in each column.

A sports coach intends to choose a team of players from a pool of candidates. The coach wants to be able to have more than 20 but fewer than 25 distinct possibilities for the composition of the chosen team, with at least as many candidates chosen for the team as those not chosen.

Identify the number of candidates in the pool and the number of players on the team that are consistent with the coach’s intentions. Make only two selections, one in each column.

A sports coach intends to choose a team of players from a pool of candidates. The coach wants to be able to have more than 20 but fewer than 25 distinct possibilities for the composition of the chosen team, with at least as many candidates chosen for the team as those not chosen.

Identify the number of candidates in the pool and the number of players on the team that are consistent with the coach’s intentions. Make only two selections, one in each column.

We can see that n = 7 as the number of candidates and k = 5 as the number of chosen players fulfill the requirements.

Is the question not saying \(nCr = nCn-r\) and find value of n and r for which \(25>nCr>20\) ??

The question says "with at least as many candidates chosen for the team as those not chosen" and in our case chosen is \(5 \geq 2\) which are not chosen.In addition, 7C5 = 7C2 = 7*6/2 = 21 > 20 and 21 < 25.And for n = 8, there is no k fulfilling the required conditions. _________________

nCk = nC(n - k) is the easiest to compute for k = 2 being equal to \(\frac{n(n-1)}{2}.\)You should start with the highest values and go backwards, as 6C3 = 20 and nCk is the highest for k around half of n.So, you can deduce that n must be greater than 6. And take advantage of the multiple choice question, as there is just one correct answer.

Otherwise, once you understand Pascal's triangle, it is quite easy to write it down. Having all the values in front of yours eyes, really helps to pinpoint the correct answer. _________________

For choosing teams, you want to keep in mind the combination basics...you want to choose out of n people, create a team of r. THe number of combinations (NOT permutations) should be between 21-24. That's more than 20 and less than 25.

So, if you try the various combinations the only one that works is out of 7 choose a team of 5.

THat's 7NCR57! / (5! 2!) = 6*7/2 = 21

What about 8C4?

8! / (4!*4!) = 5*6*7*8/4 = 30*7*2 = 240

What about 8C5?8! / (5! 3!) = 6*7*8 / (3*2) = 56

What about 8C6?8! / (6! 2!) = 7*8 / 2 = 28

What about 8C7?8! / (7! 1!) = 8

So somehow we don't get in the correct range. You can try for the others but you won't get in range.

So if you're familiar with the binomial distribution curve for these combinations and that out of 8 you choose a number in the middle you'll get the highest number of results. Using that you can do an educated guess as to what to try next. _________________

For choosing teams, you want to keep in mind the combination basics...you want to choose out of n people, create a team of r. THe number of combinations (NOT permutations) should be between 21-24. That's more than 20 and less than 25.

So, if you try the various combinations the only one that works is out of 7 choose a team of 5.

THat's 7NCR57! / (5! 2!) = 6*7/2 = 21

What about 8C4?

8! / (4!*4!) = 5*6*7*8/4 = 30*7*2 = 240

What about 8C5?8! / (5! 3!) = 6*7*8 / (3*2) = 56

What about 8C6?8! / (6! 2!) = 7*8 / 2 = 28

What about 8C7?8! / (7! 1!) = 8

So somehow we don't get in the correct range. You can try for the others but you won't get in range.

So if you're familiar with the binomial distribution curve for these combinations and that out of 8 you choose a number in the middle you'll get the highest number of results. Using that you can do an educated guess as to what to try next.

But what does the question mean by"with at least as many candidates chosen for the team as those not chosen". _________________

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